4.2.2 Case 2: Strong Signal, Strong Background, Nonoptimized Gain

If we now let $G=30$ instead of the optimized $G=59$ used above, we obtain $\rho_0 =9.95$, $\rho_+=17.6$, $\Delta=60.8$, $\beta_0=0.566$. A full simulation was run to determine $C({\mathbf a})$ and $\nabla C({\mathbf a})$ at this operating point. The capacity sensitivity with respect to the fundamental parameters are shown in Fig. 4. Again, by far, the SNR parameter $\rho_0$ has the greatest effect on capacity, followed by the excess SNR parameter $\rho_+$. And again, the blending fraction $\beta_0$ and skewness difference $\Delta$ play a lesser role.

The Jacobian for Case 2 is given in Appendix B, from which we obtain the capacity with respect to the physical parameters. The capacity sensitivity with respect to the physical parameters is shown in Fig. 5. As can be seen, capacity is sensitive in a very similar way as in case 1, except that the non-optimized gain $G$ is easily identified by its much larger value. In this case, the capacity sensitivity with respect to $G$ is about ten times as it is in case 1, which is an indication that capacity may be increased by properly increasing the gain.

Figure 4: Capacity sensitivity of $2^k$-PPM, with respect to fundamental parameters, case 2.

Figure 5: Capacity sensitivity of $2^k$-PPM with respect to physical parameters, case 2.